Length Functions and HNN Groups

Subgroups of Quasi-hnn Groups

On Groups which contain no HNN-Extensions

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 23 We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 25 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine...

Comparing Length Functions on Free Groups

The aim of this note is to study a pair of length functions on a free group, associated to a point in the unprojectivized Outer space. We discuss various characteristics which compare them and describe how these may be unified by a “Manhattan curve” analogous to those used to study pairs of hyperbolic surfaces. Our approach involves the thermodynamic formalism used in the ergodic theory of hype...

Solvable Groups of Exponential Growth and Hnn Extensions

An extraordinary theorem of Gromov, [4], characterizes the finitely generated groups of polynomial growth; a group has polynomial growth iff it is nilpotent by finite. This theorem went a long way from its roots in the class of discrete subgroups of solvable Lie groups. Wolf, [11], proved that a polycyclic group of polynomial growth is nilpotent by finite. This theorem is primarily about linear...

Ascending Hnn - Extensions and Properly 3 - Realisable Groups

The most useful constructions in Combinatorial Group Theory are amalgamated free products and HNN-extensions, and they are the two basic examples in the theory of graphs of groups due to Bass and Serre (see [9]). We recall that given a group G and a subgroup H 6 G together with monomorphisms (respectively homomorphisms) ψ, φ : H −→ G, the group determined by the presentation 〈 G, t; t−1ψ(h)t = ...